Problem: The polynomial
\[ax^4 + bx^3 + cx^2 + dx + e = 0\]has coefficients that are all integers, and has roots $-2,$ $5,$ $9,$ and $-1/3.$  If $e$ is a positive integer, then find its smallest possible value.
Explanation: By the Integer Root Theorem, $-2,$ $5,$ and $9$ must all divide $e,$ so $e$ must be at least 90.  The polynomial
\[(x + 2)(x - 5)(x - 9)(3x + 1) = 3x^4 - 35x^3 + 39x^2 + 287x + 90\]satisfies the given conditions, so the smallest possible value of $e$ is $\boxed{90}.$